Subgroups — #30649256

Order 16256.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550504	C₂	1	1	yes	maximal, characteristic
2	#22550286	C₂	1	2	yes	maximal, complemented
3	#22550282	C₂	1	2	yes	maximal, complemented
4	#22550275	C₂	1	2	yes	maximal, complemented
5	#22550272	C₂	1	2	yes	maximal, complemented
6	#22549919	C₂	1	1	yes	maximal, characteristic
7	#22549916	C₂	1	1	yes	maximal, characteristic
8	#22549912	C₂	1	1	yes	maximal, characteristic
9	C₂×#16861140	C₂	1	1	yes	maximal, characteristic
10	#22549832	C₂	1	1	yes	maximal, characteristic
11	#544×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14523232	C₂²	1	1	yes	characteristic
13	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16861209	C₂²	1	2	yes	—
15	#16861208	—	2	1	no	—
16	#16861201	C₂²	1	2	yes	—
17	#16861200	—	2	1	no	—
18	#16861186	—	2	2	no	—
19	#16861168	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
21	#16861142	C₂²	1	2	yes	—
22	#16861142	C₂²	1	2	yes	—
23	#16861142	C₂²	1	1	yes	characteristic
24	#16861142	C₂²	1	1	yes	characteristic
25	#16861140	C₂²	1	1	yes	characteristic
26	#16861140	C₂²	1	2	yes	—
27	#16861140	C₂²	1	2	yes	—
28	#16861140	C₂²	1	2	yes	—
29	#16861140	C₂²	1	1	yes	characteristic
30	#16861140	C₂²	1	2	yes	—
31	#16861120	C₂²	1	2	yes	—
32	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
34	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
35	#125×C₁₂₇	—	2	1	no	—
36	#125×C₁₂₇	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
38	#117×C₁₂₇	C₂²	1	1	yes	characteristic
39	#117×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₅₄	—	2	2	no	—
42	C₂²×Dic₁₂₇	—	2	1	no	—
43	#14523232	Q₈	1	4	yes	complemented
44	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	complemented
45	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₂₇	—	2	2	no	—
49	C₄⋊Dic₁₂₇	—	2	1	no	—
50	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
52	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
53	Dic₁₂₇⋊C₄	—	2	2	no	—
54	Dic₁₂₇⋊C₄	—	2	2	no	—
55	C₂²⋊Dic₁₂₇	—	2	1	no	—
56	C₄×Dic₁₂₇	—	2	1	no	—
57	C₂×Q₈×C₁₂₇	—	2	2	no	—
58	C₂²×C₄×C₁₂₇	—	2	1	no	—
59	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₂₇	—	2	1	no	—
61	C₄⋊C₄×C₁₂₇	—	2	2	no	—
62	C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
70	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
71	C₄²×C₁₂₇	—	2	1	no	—
72	C₄²×C₁₂₇	C₂³	1	2	yes	—
73	Dic₂₅₄	—	8	2	no	—
74	C₂×Dic₁₂₇	—	2	1	no	—
75	C₂×Dic₁₂₇	—	2	1	no	—
76	C₂×Dic₁₂₇	—	2	2	no	—
77	C₂×Dic₁₂₇	—	8	1	no	—
78	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
79	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
80	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
81	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₁₂₇	—	8	2	no	—
83	C₂×C₄×C₁₂₇	—	2	1	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	8	1	no	—
87	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
93	C₂²×C₁₂₇	—	2	1	no	—
94	C₂²×C₁₂₇	—	2	1	no	—
95	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	—	4	1	no	—
100	C₄×C₁₂₇	—	4	2	no	—
101	C₄×C₁₂₇	—	4	1	no	—
102	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
103	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
104	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
105	Dic₁₂₇	—	4	1	no	—
106	Dic₁₂₇	—	4	2	no	—
107	Dic₁₂₇	—	4	1	no	—
108	C₂×C₁₂₇	—	2	2	no	—
109	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₁₂₇	#577	1	1	yes	characteristic
111	C₂×C₁₂₇	#577	1	1	yes	characteristic
112	#3163	—	127	1	no	maximal, Sylow
113	C₁₂₇	#3163	1	1	yes	characteristic, complemented, Sylow
114	#544	—	127	1	no	—
115	#544	D₁₂₇	1	1	yes	characteristic
116	#501	—	127	2	no	—
117	#501	—	127	2	no	—
118	#499	—	127	2	no	—
119	#498	—	127	2	no	—
120	#428	—	127	1	no	—
121	#428	—	127	1	no	—
122	#427	—	127	1	no	—
123	C₂×#106	—	127	1	no	—
124	#405	—	127	1	no	—
125	C₂×M₁₆	—	127	1	no	—
126	C₂²×C₈	—	127	1	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	—	127	2	no	—
129	C₄⋊Q₈	—	254	1	no	—
130	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	—	127	2	no	—
133	#125	—	254	1	no	—
134	#125	C₂×D₁₂₇	1	2	yes	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	—	254	2	no	—
137	#117	—	127	1	no	—
138	#117	C₂×D₁₂₇	1	1	yes	characteristic
139	#117	C₂×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	127	1	no	—
141	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
142	#107	—	127	1	no	—
143	#107	—	127	1	no	—
144	#107	—	127	2	no	—
145	#107	—	127	2	no	—
146	#106	—	127	2	no	—
147	#106	—	127	2	no	—
148	#106	—	127	1	no	—
149	#106	—	127	1	no	—
150	#106	—	127	2	no	—
151	#106	—	127	2	no	—
152	#97	—	127	2	no	—
153	C₄×C₈	—	127	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	254	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	254	1	no	—
158	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
159	M₁₆	—	127	4	no	—
160	C₂×C₈	—	127	4	no	—
161	C₂×C₈	—	127	1	no	—
162	C₂×C₈	—	127	2	no	—
163	C₂×C₈	—	127	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	127	2	no	—
169	C₄⋊C₄	—	127	1	no	—
170	C₄⋊C₄	—	127	1	no	—
171	C₄⋊C₄	—	254	2	no	—
172	C₄⋊C₄	—	254	2	no	—
173	C₄⋊C₄	—	254	1	no	—
174	C₄⋊C₄	—	254	2	no	—
175	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
177	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
178	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
179	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	254	1	no	—
182	C₂²⋊C₄	C₂²⋊D₁₂₇	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	254	1	no	—
185	C₄²	C₂²×D₁₂₇	1	2	yes	—
186	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	1016	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	1016	1	no	—
194	C₂×C₄	—	254	1	no	—
195	C₂×C₄	—	254	1	no	—
196	C₂×C₄	—	254	2	no	—
197	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
200	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
201	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
202	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
203	C₈	—	127	2	no	—
204	C₈	—	127	2	no	—
205	C₈	—	127	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16861207	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	508	2	no	—
216	C₄	—	508	1	no	—
217	C₄	—	508	1	no	—
218	C₄	#16861207	1	2	yes	—
219	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#16861207	1	1	yes	characteristic
223	C₂	#22550684	1	1	yes	characteristic
224	C₂	#22550687	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#544×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#544	D₁₂₇
hypocenter	C₁₂₇	#3163
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22550286	C₂	semidirect
3. #22550282	C₂	semidirect
4. #22550275	C₂	semidirect
5. #22550272	C₂	semidirect
43. #14523232	Q₈	semidirect
44. C₂×C₁₂₇⋊C₈	Q₈	semidirect
113. C₁₂₇	#3163	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22550504	2
#22550286	2
#22550282	2
#22550275	2
#22550272	2
#22549919	2
#22549916	2
#22549912	2
C₂×#16861140	2
#22549832	2
#544×C₁₂₇	2
#3163	127